A tzaddik's true offspring are mitzvos

אלה תולדות נח נח איש צדיק פרש"י עיקר תולדותיהם של צדיקים מצות ומעשים טובים.

"These are the offspring of Noach — Noach was a righteous man." Rashi explains: the essential offspring of tzaddikim are mitzvos and good deeds.

Although the verse seems to introduce Noach's children, Rashi teaches that a tzaddik's true "offspring" are his mitzvos and ma'asim tovim — these are what he truly brings into the world.

וכ' אלה פסל אחרים שלא הי' לו תולדות אחרים רק מצות ומעש"ט שהצדיק מוליד תולדות בהתקשרות הנשמה אל השורש ונעשה הגוף טפל שזה בחי' צדיק דאחיד בשמיא וארעא.

And it is written "these," which excludes others — meaning he had no other offspring except mitzvos and good deeds; for the tzaddik begets offspring through the binding of the neshamah to its root, so that the body becomes secondary — and this is the level of the tzaddik who "grasps both heaven and earth."

The word "these" comes to limit: a tzaddik's real legacy is purely his mitzvos. He "gives birth" by binding his soul back to its source, making the body subordinate. Such a tzaddik joins the upper world and the lower world together — he holds onto both shamayim and aretz.

ז"ש צדיק הי' בדורותיו כי הי' צדיק בב' הדירות כי האדם יש לו דירה למעלה ולמטה וכשמכין לו בפרוזדור לבוא אל הטרקלין נקרא תמים בדורותיו.

This is the meaning of "he was a tzaddik in his generations (b'dorosav)" — for he was a tzaddik in his two dwellings (diros); for a person has a dwelling above and a dwelling below, and when he prepares himself in the antechamber (prozdor) in order to enter the banquet hall (traklin), he is called "tamim (perfect) in his dwellings."

The Sefas Emes reads "dorosav" as "dirosav" — dwellings. Every person lives in two homes, an upper spiritual one and a lower physical one. This world is the prozdor, the entrance hall before the traklin of the World to Come; one who uses the lower dwelling rightly to prepare for the upper is called "perfect in his dwellings."

וכן הי' התולדות שלו.

And such were his offspring.

Noach's "children" — his mitzvos — likewise carried this quality of binding the lower world to the upper.

וע"ז מבקשין ולא נלד לבהלה רק בדביקות אל השורש:

And concerning this we pray: "that we should not give birth in vain (la'behalah)," but rather in deveikus (attachment) to the root.

We ask that the "offspring" we bring forth — our deeds and even our children — not be born to confusion and emptiness, but be bound in deveikus to their Source, fulfilling their true purpose.

Summary: A tzaddik's true offspring are his mitzvos and good deeds, born by binding the neshamah to its root and making the body secondary. Such a person, like Noach, is "perfect in his dwellings" — using this world as a prozdor to enter the next — and we pray that all we produce be born not in vain but in deveikus to Hashem.